In parallelogram ABCD, two points P and Q are taken on diagonal BD such that DP = BQ (see Fig. 8.20). Show that:



(i) Δ APD ≅Δ CQB

(ii) AP = CQ

(iii) Δ AQB ≅Δ CPD

(iv) AQ = CP

(v) APCQ is a parallelogram

(i) In ΔAPD and ΔCQB,

∠ADP = ∠CBQ (Alternate interior angles for BC || AD)

AD = CB (Opposite sides of parallelogram ABCD)

DP = BQ (Given)

Two sides and included angle (SAS) Definition: Triangles are congruent if any pair of corresponding sides and their included angles are equal in both triangles.

ΔAPD ΔCQB (Using SAS congruence rule)

(ii) As we had observed that,

ΔAPD ΔCQB

AP = CQ (CPCT)

(iii) In ΔAQB and ΔCPD,

∠ABQ = ∠CDP (Alternate interior angles for AB || CD)

AB = CD (Opposite sides of parallelogram ABCD)

BQ = DP (Given)

Two sides and included angle (SAS) Definition: Triangles are congruent if any pair of corresponding sides and their included angles are equal in both triangles.

ΔAQB ΔCPD (Using SAS congruence rule)

(iv) As we had observed that,

ΔAQB ΔCPD,

AQ = CP (CPCT)

(v) From the result obtained in (ii) and (iv),

AQ = CP and AP = CQ

Since,

Opposite sides in quadrilateral APCQ are equal to each other,

APCQ is a parallelogram